Adjoint equation: Difference between revisions

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|Meaning=An equation of the form '''x'''<sub>0</sub> = &#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>, in which the [[linear operator]] &#x1D5DF;<sup>''T''</sup> is the adjoint  of the matrix [[operator]] &#x1D5DF; that satisfies (&#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>,'''x'''<sub>0</sub>) = ('''x'''<sub>1</sub>,&#x1D5DF;'''x'''<sub>0</sub>), where '''x'''<sub>0</sub> and '''x'''<sub>1</sub> are vectors and (,)  represents an [[inner product]].
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|Explanation=If (,) is the standard dot product (Euclidean inner product) then &#x1D5DF;<sup>''T''</sup> is simply the transpose of  &#x1D5DF;. <br/>''See'' [[adjoint sensitivity]], [[adjoint model]], [[tangent linear equation]].
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== adjoint equation ==
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<div class="definition"><div class="short_definition">An equation of the form '''x'''<sub>0</sub> = &#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>, in which the [[linear operator]] &#x1D5DF;<sup>''T''</sup> is the adjoint  of the matrix [[operator]] &#x1D5DF; that satisfies (&#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>,'''x'''<sub>0</sub>) = ('''x'''<sub>1</sub>,&#x1D5DF;'''x'''<sub>0</sub>), where '''x'''<sub>0</sub> and '''x'''<sub>1</sub> are vectors and (,)  represents an [[inner product]].</div><br/> <div class="paragraph">If (,) is the standard dot product (Euclidean inner product) then &#x1D5DF;<sup>''T''</sup> is simply the transpose of  &#x1D5DF;. <br/>''See'' [[adjoint sensitivity]], [[adjoint model]], [[tangent linear equation]].</div><br/> </div>
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Latest revision as of 21:17, 27 March 2024

An equation of the form x0 = 𝗟Tx1, in which the linear operator 𝗟T is the adjoint of the matrix operator 𝗟 that satisfies (𝗟Tx1,x0) = (x1,𝗟x0), where x0 and x1 are vectors and (,) represents an inner product.

If (,) is the standard dot product (Euclidean inner product) then 𝗟T is simply the transpose of 𝗟.
See adjoint sensitivity, adjoint model, tangent linear equation.

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